[Paper Review] Eigenlogic: Interpretable Quantum Observables with applications to Fuzzy Behavior of Vehicular Robots

Zeno Toffano, François Dubois|arXiv (Cornell University)|Jul 17, 2017

Quantum Computing Algorithms and Architecture7 references3 citations

TL;DR

This paper introduces Eigenlogic, a quantum-inspired logical framework where propositions are represented as quantum observables with eigenvalues as truth-values and eigenvectors as interpretations. By applying the Born rule to non-eigenvector states, fuzzy membership values emerge naturally, enabling interpretable multivalued and fuzzy logic in autonomous robotic agents, demonstrated via Braitenberg vehicles with enhanced behavioral decision-making capabilities.

ABSTRACT

This work proposes a formulation of propositional logic, named Eigenlogic, using quantum observables as propositions. The eigenvalues of these operators are the truth-values and the associated eigenvectors the interpretations of the propositional system. Fuzzy logic arises naturally when considering vectors outside the eigensystem, the fuzzy membership function is obtained by the Born rule of the logical observable.This approach is then applied in the context of quantum robots using simple behavioral agents represented by Braitenberg vehicles. Processing with non-classical logic such as multivalued logic, fuzzy logic and the quantum Eigenlogic permits to enlarge the behavior possibilities and the associated decisions of these simple agents.

Motivation & Objective

• To develop a logical framework that interprets propositional logic through quantum observables for enhanced interpretability.
• To address the limitations of classical logic in modeling uncertain or intermediate states in robotic decision-making.
• To demonstrate how fuzzy logic arises naturally from quantum formalism without ad hoc definitions.
• To apply this framework to simple autonomous agents (Braitenberg vehicles) to expand their behavioral repertoire.

Proposed method

• Represent logical propositions as Hermitian quantum observables, with eigenvalues as truth-values and eigenvectors as semantic interpretations.
• Use the Born rule to compute fuzzy membership values for states outside the eigensystem, enabling multivalued logic.
• Model behavioral agents as Braitenberg vehicles whose decisions are governed by Eigenlogic propositions.
• Construct logical operators using quantum mechanical formalism, preserving consistency with quantum probability.
• Map logical connectives to unitary transformations or observable combinations in Hilbert space.
• Ensure interpretability by anchoring truth-values and meanings in the spectral decomposition of observables.

Experimental results

Research questions

• RQ1 How can propositional logic be reformulated using quantum observables to provide both truth-values and semantic interpretations?
• RQ2 In what way does fuzzy logic emerge naturally from the quantum formalism when states are not eigenstates of the observable?
• RQ3 Can Eigenlogic enhance decision-making in simple autonomous agents like Braitenberg vehicles beyond classical logic?
• RQ4 How does the interpretability of truth-values and meanings improve the transparency of robotic behavior?
• RQ5 What is the role of the Born rule in generating fuzzy membership functions within this logical framework?

Key findings

• Fuzzy membership values are derived naturally from the Born rule applied to non-eigenstates, eliminating the need for ad hoc definitions.
• The framework provides a unified representation where truth-values and semantic interpretations are intrinsically linked via eigenvectors.
• Braitenberg vehicles equipped with Eigenlogic exhibit richer, more nuanced behaviors than those using classical logic.
• The use of quantum observables enables multivalued logic without compromising interpretability.
• The method supports transparent, semantic-aware decision-making in autonomous agents through quantum-inspired logic.
• The approach demonstrates that quantum formalism can be used to model fuzzy reasoning in a physically and mathematically coherent way.